The calculations look good, but the statements surrounding it do not. In particular:
1. It doesn't make sense to say that you choose m such that some statement is true for all m.
2. You didn't say that ##\varepsilon## is a positive real number, so the reader can wonder how you intend to make ##|x_n-x_m|<\varepsilon## when ##\varepsilon=-1##.
3. The calculation didn't actually conclude that ##|x_n-x_m|<\varepsilon##.
Edit: 4. If the [x] notation means what I think it does, your definition of N doesn't work.